# Properties of MaxEnt posterior distribution for a die with prescribed average

Question: after throwing a die a large number of times and discovering that the average of the outcomes is $4$, what probability distribution one should assign to statements "the next roll will be $i$" for $i = 1, 2, \dots, 6$?

E.T. Jaynes in Chapter 9 of the book "Probability Theory: The Logic of Science" derives the following:

if we start with ignorance knowledge, $I_0$, meaning that all individual rolls are independent and are equally likely to occur, so $1/6$ for each outcome, then we just need to find $(p_1, p_2, \dots, p_6)$ that maximises $H(p_1, p_2, \dots, p_6) := \sum_{i=1}^6 -p_i \log p_i$ subject to $\sum_{i=1}^6 i p_i = 4$.

Utilising Lagrange multipliers one can easily derive Boltzmann posterior distribution for such a die. My posterior distribution, found numerically, is the following:

$$(p_1, p_2, p_3, p_4, p_5, p_6) \approx (0.10,0.12,0.15,0.17,0.21,0.25).$$

Moreover E.T. Jaynes advocates that such posterior distribution is the only answer consistent with prior knowledge $I_0$, the data $D=\{\text{the mean is$4$}\}$ and Cox's theorem. However, I have a few questions about such posterior:

1) Qualitative: does it really do what common sense dictates it should do? Why the posterior doesn't have more mass on $4$?

2) Why the mode of the posterior is $6$ rather than $4$? Under what loss function should I guess 4?

3) Why the MLE approach fails to give the mode of the posterior distribution, despite the following quote:

A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters. (Wiki: MLE).

l = concat $map (\a -> [(-a), a]) l' where l' = map (/ 1000) [1.. ] findExp xs = sum$ zipWith (*) [1..] xs